How to create a voronoi texture with different color in every tile and black lines in between?
To make all the tiles different color you could just take the color output of the voronoi texture and play with that. It's already quite close to what you want. If you want the black lines you need to add more nodes to mix it with.

Update
Here are two different results. This one has uniform lines and the last one has little distortion added so it closer to the original picture.